The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces
Karl‐Theodor Sturm
Abstract
Equipped with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 2 comma q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{2,q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -distortion distance <inline-formula content-type="math/tex"> <tex-math>\DD _{2,q}</tex-math> </inline-formula> , the space <inline-formula content-type="math/tex"> <tex-math>\XX _{2q}</tex-math> </inline-formula> of all metric measure spaces <inline-formula content-type="math/tex"> <tex-math>(X,\d ,\m )</tex-math> </inline-formula> is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on <inline-formula content-type="math/tex"> <tex-math>\ol \XX _{2q}</tex-math> </inline-formula> are presented.